Our strategy to prove the robust radical SG theorem is based on the following toy example: suppose our polynomial ring is \([x_1, \ldots , x_r, y_1, \ldots , y_s]\) , where one should think of \(s\)  being constant and \(r \gg s\) , and every quadratic form \(Q\)  in our configuration is a polynomial which is “univariate” over the smaller polynomial ring \([y_1, \ldots , y_s]\) .
That is, for each quadratic \(Q\) , there exists a linear form \(x_Q \in {x_1, \ldots , x_r}\)  such that \(Q \in [x_Q, y_1, \ldots , y_s]\) .
In this case, one would hope that the non-linear SG dependencies involving our configuration \(\)  would imply linear SG dependencies for the set of linear forms \(_1 \cup \lbrace  x_Q \ \mid \ Q \in _2 \rbrace \) .
If we manage to prove that the latter set of linear forms is a robust linear SG configuration, we can invoke the robust SG theorem for linear forms of [1]}, [2]}.
